Problem Solving

A marketing firm determined that of 200 households surveyed 80 used neither Brand A nor Brand B soap, 60 used only Brand A soap, and for every household that used both brands of soap, 3 used only brand B soap, how many of the 200 households surveyed used both brands of soap?

15


Can someone explain why the double matrix is not working for this problem?

Out of 200 , 80 uses nether A or B
so no of user using brand A or B = 120
if "x" is the num of house hold using both brand then "3x" will be the # of user using brand B
60+x+3x = 120 (apply set theory)
==> x = 15

amitabhprasad wrote:Out of 200 , 80 uses nether A or B
so no of user using brand A or B = 120
if "x" is the num of house hold using both brand then "3x" will be the # of user using brand B
60+x+3x = 120 (apply set theory)
==> x = 15

According to the set theory:

Total = (a) + (b) -both+ none

200 =60 + 3x -x +80
x= 30

where I went wrong ?

[quote="Bidisha800"][quote="amitabhprasad"]Out of 200 , 80 uses nether A or B
so no of user using brand A or B = 120
if "x" is the num of house hold using both brand then "3x" will be the # of user using brand B
60+x+3x = 120 (apply set theory)
==> x = 15[/quote]

According to the set theory:

Total = (a) + (b) -both+ none

200 =60 + 3x -x +80
x= 30

where I went wrong ?[/quote]


It's not Total = (a) + (b) - both + none.
The correct formula is Total = (a) + (b) + both + none and then you get the right answer of x = 15. Notice that you do not have "at least" anywhere in the problem, therefore your formula is not correct.

Bidisha800 wrote:

amitabhprasad wrote:Out of 200 , 80 uses nether A or B
so no of user using brand A or B = 120
if "x" is the num of house hold using both brand then "3x" will be the # of user using brand B
60+x+3x = 120 (apply set theory)
==> x = 15

According to the set theory:

Total = (a) + (b) -both+ none

200 =60 + 3x -x +80
x= 30

where I went wrong ?

very common mistake we make

a U b = a + b - a <intersection> b

but here a is (total a) that is (only a + both a and b )

Double Matrix soln

A Not A Total

B x 3x 60

Not B 60 80 140

total x+60 3x+80 200


solve for x.

Explanation for how I solved this one:
Notice that any given household has 4 options:
- they use both brands of soap - x
- they use only brand A - 60 hh
- they use only brand B - 3x
- they don't use any of the two - 80 hh
Since you do not see "at least" anywhere in the problem, then adding the number of household that picked either one of the four would give you the sum of 200 households that participated in the survey. So you will have:

200 = 3x + x + 60 + 80, so 4x = 60, x = 15.

Be careful though: if "at least" appears anywhere in the problem, that changes thing quite a bit. Because if a hh says it uses at least brand B of soap, it could use only brand B, but it could also use both brands.

Folks, you can avoid confusion if you can use a simple venn diagram.

Have a look at this....



Once you master Venn diagrams I think you can avoid drawing them at least in the case where there are two variables.

How do you know the ratio is 1 to 3 though? The problem says that "for every household that used both brands of soap , 3 used only brand B"

So doesn't this mean that out of all the households that used both A and B, 3 only used brand B?

hutch27 wrote:How do you know the ratio is 1 to 3 though? The problem says that "for every household that used both brands of soap , 3 used only brand B"

So doesn't this mean that out of all the households that used both A and B, 3 only used brand B?

The phrase in red is incorrect.
For every X, there are 3 Y's.
This means that for every ONE element that is an X, THREE elements are Y's.
In other words:
X:Y = 1:3.

In the problem above:
For every household that used both brands of soap, 3 used only brand B.
In math terms:
(both soaps) : (only B) = 1:3.